Local homology

The homology groups (cf. Homology group)

defined at points , where is homology with compact support. These groups coincide with the direct limits

over open neighbourhoods of , and for homologically locally connected they also coincide with the inverse limits

The homological dimension of a finite-dimensional metrizable locally compact space over (cf. Homological dimension of a space) coincides with the largest value of for which , and the set of such points has dimension .

Let be the differential sheaf over defined by associating with each open set the chain complex . The groups are the fibres of the derived sheaves . For generalized manifolds, for . In this case the homology sequence of the pair with coefficients in coincides with the cohomology of the pair with coefficients in the sheaf (Poincaré–Lefschetz duality). The similar facts for the local cohomology of locally compact spaces do not hold.

References

[1]	E.G. Sklyarenko, "On the theory of generalized manifolds" Math. USSR Izv. , 5 : 4 (1971) pp. 845–858 Izv. Akad. Nauk SSSR Ser. Mat. , 35 (1971) pp. 831–843
[2]	A.E. [A.E. Kharlap] Harlap, "Local homology and cohomology, homology dimension and generalized manifolds" Math. USSR Sb. , 25 : 3 (1975) pp. 323–349 Mat. Sb. , 96 (1975) pp. 347–373

References

[a1]	A. Dold, "Lectures on algebraic topology" , Springer (1972) pp. Sect. IV.3